Memory Test Failure

I ran the Dell diagnostic test on my Dell Dimension 8400 and got the following failures:

System Memory - Data Bus Stress

System Memory - MATS

System Memory - March A Test

System Memory - March B Test

System Memory - March C Test

System Memory - March X Test

System Memory - March Y Test

I received the following errors:

2F00:0B13

2F2F:0119

2F2F:0419

2F2F:0819

2F2F:0719

2F2F:0919

2F2F:0A19

2F2F:0B1C

Does this mean that one or more of my computer's hardware is faulty? If so, is the only way to fix this problem to replace the relevant parts? My computer's warranty has expired, so do I have to renew it?

Certain programs constantly crash unexpectedly (eg firefox), almost no programs can be installed due to some corruption (even from reputable sites like CNET), and many programs are unable to function properly eg Avast's on-access protection cannot be turned on. I have already checked using Antivirus software and Hijackthis, and no viruses were found. Also, I occasionally get a BSOD citing some problem with the NTFS.

Trying to conduct a system restore failed, and so I ran the diagnostics under that symptom.